SE has created 3 Methods of Summoning in Final Fantasy
Partner Summoning - Summon Summoned Monsters as an ally to fight by the player size which is the method of Summoning most people know.
Society Summoning- The Summoning Method Beasts Tribes uses to Summon their God
Incarnate Summoning- The Summoning method used in FFXIII by Ciaus to when he manifested Bahumat in his own body transforming into the Eidolon. Louisoix Leveilleur used Incarnate Summoning to Summon the Primal Pheonix and Lady Iceheart uses Incarnate Summoning to Summon Shiva.

Incarnate Summoning is an ability that can be given to the Summoner and work similar to WoW Warlock Metamorphosis where the player transforms into the Demon temporaraly with a 10s cool down.